2020-05-04T09:54:34Z
Acceptance in OA@INAF
Lower limit for differential rotation in members of young loose stellar associations
Title
DISTEFANO, Elisa Maria Carmela; Lanzafame, Alessandro Carmelo; LANZA,
Antonino Francesco; MESSINA, Sergio; Spada, F.
Authors
10.1051/0004-6361/201527698
DOI
http://hdl.handle.net/20.500.12386/24417
Handle
ASTRONOMY & ASTROPHYSICS
Journal
591
Number
DOI:10.1051/0004-6361/201527698 c ESO 2016
Astronomy
&
Astrophysics
Lower limit for differential rotation in members of young loose
stellar associations
?
E. Distefano
1, A. C. Lanzafame
2, A. F. Lanza
1, S. Messina
1, and F. Spada
31 INAF–Osservatorio Astrofisico di Catania, via S. Sofia, 78, 95123 Catania, Italy e-mail: elisa.distefano@oact.inaf.it
2 University of Catania, Astrophysics Section, Dept. of Physics and Astronomy, via S. Sofia, 78, 95123 Catania, Italy 3 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
Received 4 November 2015/ Accepted 5 April 2016
ABSTRACT
Context.Surface differential rotation (SDR) plays a key role in dynamo models and determines a lower limit on the accuracy of stellar rotation period measurements. SDR estimates are therefore essential to constrain theoretical models and infer realistic rotation period uncertainties.
Aims.We measure a lower limit to SDR in a sample of solar-like stars belonging to young loose stellar associations with the aim of investigating how SDR depends on global stellar parameters in the age range (4−95 Myr).
Methods.The rotation period of a solar-like star can be recovered by analyzing the flux modulation caused by dark spots and stellar rotation. The SDR and the latitude migration of dark-spots induce a modulation of the detected rotation period. We employed long-term photometry to measure the amplitude of such a modulation and to compute the quantity∆Ωphot= P2π
min−
2π
Pmaxthat is a lower limit
to SDR.
Results.We find that∆Ωphotincreases with the stellar effective temperature and with the global convective turn-over timescale τc, which is the characteristic time for the rise of a convective element through the stellar convection zone. We find that ∆Ωphot is proportional to T2.18±0.65
eff in stars recently settled on the ZAMS. This power law is less steep than those found by previous authors, but closest to recent theoretical models. We investigate how∆Ωphotchanges in time in a ∼1 M star. We find that∆Ωphotsteeply increases between 4 and 30 Myr and that it is almost constant between 30 and 95 Myr. We find also that the relative shear increases with the Rossby number Ro. Although our results are qualitatively in agreement with hydrodynamical mean-field models, our measurements are systematically higher than the values predicted by these models. The discrepancy between∆Ωphotmeasurements and theoretical models is particularly large in stars with periods between 0.7 and 2 d. Such a discrepancy, together with the anomalous SDR measured by other authors for HD 171488 (rotating in 1.31 d), suggests that the rotation period could influence SDR more than predicted by the models.
Key words. stars: solar-type – starspots – stars: rotation – open clusters and associations: general – techniques: photometric
1. Introduction
Surface differential rotation (SDR) has been detected and mea-sured in a wide sample of solar-like stars (e.g., Barnes et al. 2005;Collier Cameron 2007;Reinhold et al. 2013). These mea-surements are essential to constrain theoretical models that try to investigate the connections between rotation, convection, and the topology of stellar magnetic fields. The amplitude of SDR influ-ences the type of dynamo operating inside the star. Differential rotation is able to generate a strong toroidal magnetic field out of a weak poloidal field, and is therefore a key ingredient of αΩ dynamos (e.g.,Parker 1955;Brandenburg et al. 1991). On the other hand, stars characterized by a low SDR degree that rotate as solid bodies can instead sustain only an α2-type dynamo (e.g., Moss & Brandenburg 1995;Küker & Rüdiger 2011).
The amplitude of the SDR is usually measured through the surface rotational shear:
∆Ω = Ω0−Ωpole, (1)
? Tables 2–5 are only available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr(130.79.128.5) or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/591/A43
whereΩ0andΩpoleare the surface angular velocities at the
equa-tor and the poles, respectively. If∆Ω > 0, then the star has a solar-like SDR, meaning that the equator rotates faster than the poles. If∆Ω < 0, the star exhibits an antisolar SDR.
In recent years, several authors investigated how the SDR de-pends on stellar effective temperature and rotation period.Barnes et al. (2005)reported∆Ω measurements obtained by means of the Doppler imaging technique for a sample of ten young late-type stars. They found that∆Ω is strongly dependent on the stel-lar temperature (∆Ω ∝ Te8.93±0.31ff ) but poorly correlated to Ω0
(∆Ω ∝ Ω0.15±0.1
0 ). A strong correlation between∆Ω and Teff was
also found byCollier Cameron (2007), who enlarged the sample of stars studied byBarnes et al. (2005) and found the relation ∆Ω ∝ T8.6
eff.
The results found by these works are in quantitative dis-agreement with theoretical models developed byKitchatinov & Rüdiger (1999),Küker & Rüdiger (2011), and byKüker et al. (2011). These theoretical works predict that the SDR increases with the stellar temperature and that it is almost independent of the stellar rotation period. However, the power law expected from these models is∆Ω ∝ Teff
5500 K 2 in the range 3800−6000 K and∆Ω ∝ Teff 5500 K 20
Table 1. Stellar associations.
Association N. age Ref.
(Myr) Chamaleontis ( Cha) 9 3–5 1, 2 η Chamaleontis (η Cha) 4 6–10 1, 3, 4 TW Hydrae (TWA) 9 8–12 5, 6 β Pictoris (β Pic) 18 12–22 7, 8 Octans (Oct) 1 20–40 9, 10 Columba (Col) 14 20–40 11, 12 η Carinae (η Car) 12 20–40 11, 12
Tucana-Horologium (Tuc/Hor) 17 20–40 11, 12
Argus (Arg) 10 30–50 9, 13, 14
IC 2391 3 30–50 13, 14
AB Doradus (AB Dor) 14 70–120 15, 16, 17
Notes. For each association we report the number of members for which we estimated∆Ωphot, the age estimate, and the age reference.
References. (1) Murphy et al. (2013); (2) Feigelson et al. (2003); (3)Mamajek et al. (1999); (4)Lawson & Feigelson (2001); (5)Webb et al. (1999); (6) Barrado Y Navascués (2006); (7) Song et al. (2003); (8)Makarov (2007); (9)Torres et al. (2008); (10)Murphy & Lawson (2015); (11) Torres et al. (2001); (12) Zuckerman & Webb (2000); (13) De Silva et al. (2013); (14) Barrado y Navascués et al. (2004); (15)Mentuch et al. (2008); (16)López-Santiago et al. (2006); (17)Luhman et al. (2005).
Reinhold et al. (2013) analyzed the Quarter 3 (Q3) long-cadence photometric data collected by the Kepler mission
(Borucki et al. 2010) and searched for the SDR in a sample of about 20 000 main-sequence late-type stars. In contrast with
Barnes et al. (2005)andCollier Cameron (2007), they found that the SDR exhibits a weak dependence on temperature in the range 3800−6000 K, and that above 6000 K, it steeply increases toward higher temperatures.Reinhold et al. (2013)claimed that their re-sults, although quite scattered, agree better with the model of
Küker & Rüdiger (2011). We note, however, that these authors have no information on the ages of their targets. This lack of in-formation makes it difficult to compare their results with those of Barnes et al. (2005) andCollier Cameron (2007). The tar-gets investigated by Barnes et al. (2005)andCollier Cameron (2007)comprise pre-main-sequence stars and stars recently set-tled on the zero-age main sequence (ZAMS), while the sample of Reinhold et al. (2013) included field stars of different ages. Moreover, their data do not permit studying the SDR as a func-tion of the stellar age.
The present work is focused on a sample of 111 late-type stars belonging to 11 nearby young loose associations with known ages (Table1). The ages of these associations span the interval (4−95 Myr), and therefore our study concerns the SDR evolution during the transition between the pre-main-sequence and the main-sequence phase. We searched for the SDR by an-alyzing long-term photometric time-series collected by the All Sky Automatic Survey(ASAS, Pojmanski 1997)and the Wide Angle Search for Planets(SuperWASP, Pollacco et al. 2006).
The paper is organized in the following way: in Sect. 2 we describe the photometric data. In Sect. 3 we explain the method we employed to measure the quantity∆Ωphot1, that is, a lower
limit for the SDR. In Sect. 4 we investigate how∆Ωphotdepends
1 The subscript stands for photometric and indicates the type of data used to measure the SDR.
on global stellar parameters. In Sect. 5 we discuss the main re-sults of our study, and in Sect. 6 the conclusions are drawn.
2. Data
Messina et al. (2010, 2011) analyzed ASAS and SuperWASP time series for about 300 stars belonging to nearby young as-sociations and measured the rotation period for most of these stars. In this work we reprocessed 99 ASAS time series and 31 SuperWASP time series with a new technique (see Sect. 3) for estimating a lower limit∆Ωphotfor the rotational shear. In Table 2
we list the 111 stars studied here together with their color in-dexes and spectral types. These targets were selected from those studied inMessina et al. (2010, 2011)by picking up those for which the largest number of photometric measurements is avail-able. For nine of these targets both ASAS and SuperWASP time series are available. We note that the ASAS time series pro-cessed here span an interval time longer than those propro-cessed inMessina et al. (2010, 2011).
2.1. ASAS photometry
The ASAS time series analyzed in the present work are con-tained in the ASAS-3 photometric V-band catalog2. These time series cover a time span of about 13 yr (from 1997 to 2010) and have a typical photometric error of 0.02 mag. This error al-lows the detection of rotational modulation in young solar-like stars where the amplitude modulation ranges between 0.05 and 0.3 mag(Messina et al. 2011). In our analysis we used filtered time series obtained by selecting only the best photometric data. The time-series sampling depends on the star coordinates. We divided the ASAS time series into three groups for which the average interval tmeanbetween two consecutive observations was
'1, 2, and 3 d, respectively.
2.2. SuperWASP photometry
The SuperWASP time series were downloaded from the first public data archive(Butters et al. 2010). These time series cover a time span of about four years (from 2004 to 2008). Their time coverage is quite irregular and depends on the sky coordinates. The time series processed in the present work are typically made of two or three distinct observational seasons covering 60−300 d intervals. During each season, a target was observed every day for a few hours with a sampling rate of about ten minutes in the most favorable sky positions. The SuperWASP data therefore have a better sensitivity to periods ≤1 d than the ASAS data. Our analysis is based on the processed flux measurements ob-tained through applying the SYSREM algorithm(Tamuz et al. 2005). SuperWASP observations were collected in 2004 without any light filter. Starting from 2006, they were collected through a wide-band filter in the range 400−700 nm. Owing to di ffer-ences in spectral bands with respect to the standard Johnson V ASAS data, we analyzed the SuperWASP data independently without merging them with ASAS data.
3. Method
The surface of a solar-like star is covered by magnetically active regions (ARs) associated with dark spots and bright faculae. If the star is a differential rotator, the ARs will rotate with different 2 http://www.astrouw.edu.pl/asas/
frequencies, depending on their latitudes, and will modulate the optical flux from the star. ARs migration in latitude induces a modulation of the measured photometric period that can be used to estimate the amplitude of SDR. Based on this, the method we employed to measure the SDR on a given star consists of the followings steps:
– we segmented the photometric time-series with a sliding-window algorithm;
– we searched for the stellar rotation period in each segment; – we selected the longest and shortest detected periods and
computed the quantity ∆Ωphot = Ωmax −Ωmin, where
Ωmax=P2πmin andΩmin= P2πmax.
We assumed that Pmax and Pmin are related to time-series
seg-ments where the stellar surface is dominated by high-latitude and low-latitude ARs, respectively (or by the opposite configuration for an antisolar SDR). Since the Pmaxand Pminthus found do not
necessarily sample the whole latitude range,∆Ωphotis obviously
a lower limit for the rotational shear and therefore∆Ωphot≤∆Ω.
3.1. Time-series segmentation
Time-series segmentation is very useful in dealing with magnet-ically active stars. In fact, the typical light curves of these vari-ables are not very regular because the flux variation that is due to rotational modulation can be masked by the intrinsic evolution of ARs.
In the Sun, ARs evolve on a typical timescale τAR ' 60 d (Lanza et al. 2003). This timescale, which is about twice the so-lar rotation period, and the simultaneous occurrence of different ARs over the solar disk might in principle destroy the coher-ence of the rotational signal and prevent the detection of the rotation period. However, ARs in the Sun tend to form at ac-tive longitudes, that is, at heliographic longitudes characterized by the frequent localized emergence of new magnetic flux. These AR complexes (ARCs), forming at active longitudes, evolve on a typical timescale τARC' 200−250 d(Lanza et al. 2003). Hence,
the rotational signal is approximately coherent on time intervals of the same order as τARC.Lanza et al. (2004)analyzed the total
solar irradiance (TSI) time series collected by the SoHo satellite and showed that the use of period-search algorithms to detect the rotation period of the Sun is only possible if the long-term time series is divided into sub-series with a time extension no longer than 150 d in order to limit the effect of the evolution of ARCs.
The timescales of ARs and ARCs evolution in late-type stars other than the Sun have been studied to a limited extent because of the lack of multi-year continuous photometric time-series.
Donahue et al. (1997a,b) estimated the timescales of AR evo-lution in some tens of late-type stars by analyzing long-term time series of photometric fluxes in two 0.1 nm passbands cen-tered on the Ca II H and K emission lines. They showed that the older and less magnetically active stars tend to be AR evolution-dominated. This means that the evolution of ARs and ARCs take place on timescales comparable with the stellar rotation pe-riod. The light curves of younger and more active stars, con-versely, exhibit a pattern that remains stable for several consec-utive rotations. The typical timescales measured for these stars are τAR = 50 d and τARC = 1 yr. Similar values are reported by Hussain (2002).Messina & Guinan (2003)analyzed long-term photometric time-series collected in the Johnson V-band for six young solar analogs. They found that two of these stars are evolution-dominated, while the others have τARCvalues ranging
between 160 d and 500 d, remarkably longer than their rotation periods.
The advantage of using time-series segments is that the data exhibit a more stable pattern of flux variations than the whole time-series, but they also contain fewer points. Therefore, the ro-tation period retrieved by analyzing a time-series segment could have a low statistical significance. A compromise has been found between the length of the segments and the number of points included in the segments by adopting a length T = 50 d. For ASAS data, the average number of points per segment ranges between 16 and 50, depending on the source coordinate. For Super-WASP data, the average number of points per segment ranges between 1000 and 3000. We note that T= 50 d is also the typical τARtimescale found byDonahue et al. (1997a)for young
active stars. In Sect. 3.5 we discuss the effect of the choosing different segment durations on ∆Ωphotmeasurements.
We segmented the time series by using a sliding-window al-gorithm as sketched in Fig.1. This algorithm performs a cycle on the whole time-series. For each observation time ti, a segment
is generated by selecting all points in the time interval [ti, ti+ T],
where T is the length of the window. A segment is rejected if it has fewer then ten points or if it is a subset of another segment (this last case occurs when the time series exhibits gaps larger than the sliding window; see Fig.1).
In Fig. 1 we show the V-band time series of the star ASAS J070030-7941.8. The observations span about nine years. The red and green bullets mark the data processed byMessina et al. (2010). The different colors are used to display the segmen-tation performed in that work. The black crosses mark data col-lected after February 2008 that were not processed byMessina et al. (2010).Messina et al. (2010)extracted ten sub-series. Our sliding window algorithm allows the extraction of more seg-ments.
3.2. Rotation period and SDR estimate
We performed the period search by means of the Lomb-Scargle periodogram, which has been found to be a highly efficient method for solar-like variables(Distefano et al. 2012). For each time-series segment we computed the periodogram and selected the period P with the highest power Z. We built the distribution of the detected periods and estimated the average stellar rotation period by taking the mode of the distribution.
In the left panel of Fig.2we show the frequency histogram obtained for the source ASAS J070030-7941.8. The distribution exhibits a well-defined peak at Prot = 5.12 d that is taken as
the average stellar rotation period. The other peaks occur at the beat periods B because of the interference between the rotational modulation and the typical ASAS one-day sampling and are given by the relationship B= Prot/ |1 ± nProt| with n= 1, 2, 3, . . .
After estimating the average rotation period, we again per-formed the period search by rejecting peaks close to the beat frequencies and by selecting for each segment the highest sig-nificant peak closest to the average rotation period. We built the true period distribution and picked up Pmin and Pmax to
com-pute∆Ωphot. In some cases the period distributions show
out-liers. These outliers might be related to real rotation frequen-cies or to false positives and might lead to an overestimate of ∆Ωphot. To avoid this, we estimated Pminand Pmaxby taking the
5th and 95th percentile of the distribution. We note that this re-jection criterion is very cautious. Fewer than 1−2% of the mea-surements are distant more than 3σ from the average period for most of our targets. In the right panel of Fig. 2 we show the true period distribution obtained for ASAS J070030-7941.8 after beat rejection. We also report the error bars associated with Pmin and Pmax. In Figs.3−5we also plot the distributions
9.8 9.9 10 10.1 10.2 10.3 2000 2500 3000 3500 4000 4500 5000 V mag HJD -2450000 (days) ASAS 070030-7941.8 Sliding Window
Fig. 1.Typical V-band time series collected by the ASAS survey. The red and green bullets mark the segmentation performed byMessina et al. (2010). The black continuous boxes depict two of the extracted segments, while the dotted box indicates one of the sub-series rejected by our segmentation algorithm. The black crosses mark data that have not been processed inMessina et al. (2010). We employed a 50-day sliding window, while in the plot we sketch a 200-day sliding window for clarity.
0 20 40 60 80 100 120 1 2 3 4 5 6 Nr. of segments Detected periods ASAS 070030-7941.8 BEATS <PRot> 0 5 10 15 20 25 30 35 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 Nr. of segments Detected periods ASAS 070030-7941.8 5th PCTL 95th PCTL Error bars
Fig. 2.Left panel: distribution of detected periods for the star ASAS J070030-7941.8. The black continuous arrow indicates the average rotation period hProti. The black dotted arrows show the beat periods. Right panel: period distribution after beat rejections. Pminand Pmaxare estimated by taking the 5th and the 95th percentile of the distribution to ensure that the∆Ωphotcomputation is not driven by outliers.
obtained for ASAS J072851-3014.8, ASAS J084229-7903.9, and SWASP1 J101828.70-315002.8, which were taken as rep-resentative examples of our targets.
The total number of segments that could be used to measure the rotation period is different for each target. This number is a complex function of the time-series sampling and the stellar rotation period. We divided our data into three quality groups according to the number of segments N in which a period with FAP< 0.01 is measured. Group A includes stars with N > 200,
group B stars with N between 100 and 200, and group C stars with N < 100.
3.3. False-alarm probability estimate
We computed the false-alarm probability associated with a given period P by using the Schwarzenberg-Czerny (1998)
β distribution:
0 20 40 60 80 100 120 140 0 0.5 1 1.5 2 2.5 3 Nr. of segments Detected periods ASAS 072851-3014.8 BEAT <PRot> BEAT 0 10 20 30 40 50 1.62 1.625 1.63 1.635 1.64 1.645 1.65 1.655 1.66 Nr. of segments Detected periods ASAS 072851-3014.8 5th PCTL 95th PCTL <PRot> Error bars
Fig. 3.Same as Fig.2for ASAS J072851-3014.8.
0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 Nr. of segments Detected periods ASAS 084229-7903.9 BEATS <PRot> 0 10 20 30 40 50 60 70 6 8 10 12 14 Nr. of segments Detected periods ASAS 084229-7903.9 5th PCTL 95th PCTL
<PRot> Error bars
Fig. 4.Same as Fig.2for ASAS J084229-7903.0.
where Z is the power related to P in the periodogram, N is the number of data points, and M the number of indepen-dent frequencies, that is, the number of frequencies at which the periodogram powers are independent random variables. Unfortunately, for an unevenly sampled time series, M cannot be theoretically derived. We used the value estimated byDistefano et al. (2012),
M = 1.8Nf/k, (3)
where Nfis the number of inspected frequencies and k is a factor
introduced to take the periodogram oversampling into account. This is defined as k = (1/T)/δν, where T is the time interval spanned by the segment and δν = 0.0001 d−1is the frequency
step adopted to sample the periodogram. Equation (3) was de-rived byDistefano et al. (2012)by fitting empirical distributions of the peak powers Z as generated by Monte Carlo simulations. In our analysis we used only periods with a FAP ≤ 0.01. 3.4. Error estimate
The error associated with∆Ωphotis given by
σ2 ∆Ωphot = σ 2 Ωmax+ σ 2 Ωmin, (4)
where σΩmax = 2πσνmaxand σΩmin= 2πσνmin.
The error associated with the frequency detected by the Lomb-Scargle periodogram has two components. One
0 50 100 150 200 250 300 350 400 450 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Nr. of segments Detected periods SWASP1 J101828.70-315002.8 BEAT <PRot> 2*PRot 0 50 100 150 200 0.54 0.5405 0.541 0.5415 0.542 0.5425 0.543 Nr. of segments Detected periods SWASP1 J101828.70-315002.8 5th PCTL 95th PCTL <PRot> Error bars
Fig. 5.Same as Fig.2for SWASP1 J101828.70-315002.8.
component is due to the limited and discrete sampling of the signal, while the other is due to the data noise.Kovacs (1981) es-timated the error δνsampintroduced by the sampling for a noise–
free sinusoidal signal with frequency ν0and found that this error
is inversely correlated with ν0T2, where T is the interval spanned
by the time series.Gilliland & Fisher (1985)showed that the re-sult found byKovacs (1981)can be approximated by
δνsamp=
0.16 √
2ν0T2
· (5)
Kovacs (1981)derived this error for a uniformly sampled signal, but the result can also be applied to unevenly sampled time se-ries.Kovacs (1981)also estimated the error caused by Gaussian white noise as δνnoise= 3σ 4 √ Nν0T A , (6)
where σ is the standard deviation of the noise, N the num-ber of time-series points, and A the amplitude of the signal. Equation (6) has been derived for white Gaussian noise and can lead to an underestimation of the true error. To estimate the error caused by data noise more realistically, we used the following approach:
– we fit the data with a sinusoid;
– we computed the fit residuals to estimate the data noise; – we computed 10 000 permutations of the residuals and
con-structed 10 000 synthetic time series by adding the permuted residual to the sinusoid;
– we ran the period search algorithm on each synthetic time series; and
– we built the distribution of the detected frequencies and took the standard deviation of this distribution as an error estimate.
The error δνsamp increases with the stellar rotation period (see
Eq. (5)). It is very small and ranges from 10−4rad d−1for stars
rotating in 0.5 d to 0.006 rad d−1for stars rotating in 20 d (this is the highest rotation period detected in our target stars).
The error δνnoiseinferred from simulating synthetic time
se-ries depends on different factors such as stellar magnitude, vari-ability amplitude, and the number of observations. It ranges be-tween 0.003 rad d−1and 0.02 rad d−1.
The final errors on ∆Ωphot measurements range between
0.005 and 0.025 rad d−1.
3.5. Output of the analysis procedure
In Figs. 6−9 we show the typical output of our anal-ysis for ASAS J070030-7941.8, ASAS J072851-3014.8, ASAS J084229-7903.9, and SWASP1 J101828.70-315002.8
In the top panels we plot the rotation periods detected in the different segments vs. the mid-observation times of the same segments. In the central panels we plot the interquartile range (IQR) of the observed magnitudes, which can be regarded as an index of the variability amplitude, and in the bottom panels we plot the median magnitude. The solid lines were obtained by fit-ting the data with smoothing cubic splines and are plotted to guide the eye. The period and IQR seem to follow defined pat-terns well, which suggests that there are cyclical changes in the AR configurations. These cyclical patterns will be investigated in a forthcoming work. In some segments, no significant periods were detected. This can be ascribed to different reasons:
– the amplitude of the rotational modulation is on the same order as the photometric error and therefore is too low to be detected, or
– the AR evolution occurs on a timescale similar to or shorter than the length of the sliding window and therefore masks the variability that is due to rotational modulation.
In Figs.6−9we also plot the results obtained by using 100-d and 150-d sliding windows for comparison. In principle, the use of the longer windows should give more precise measurements of the stellar rotation period, as discussed in Sect. 3.4. In fact, the
9.95 10 10.05 10.1 10.15 10.2 10.25 10.3 10.35 2000 2500 3000 3500 4000 4500 5000 Median magnitude HJD-2450000 (days) -0.15 -0.1 -0.05 0 0.05 0.1 0.15 IQR (mag) 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 Period (days) ASAS 070030-7941.8
Fig. 6.Output of our analysis for ASAS ID J070030-7941.8. The filled circles, empty circles, and crosses show the results obtained with 50-d, 100-d, and 150-d sliding windows, respectively. The results obtained with the 100-d and 150-d sliding window have been vertically shifted to make the plot clearer. Top panel: rotation periods found in the different segments vs. time. Each point is located at the mid-time of the corresponding segment. Middle panel: IQR measured in the different segments vs. time. Bottom panel: median magnitude vs. time. The solid lines were obtained by fitting the data with smoothing cubic splines.
9.9 9.95 10 10.05 10.1 10.15 10.2 10.25 2000 2500 3000 3500 4000 4500 5000 Median magnitude HJD-2450000 (days) -0.15 -0.1 -0.05 0 0.05 0.1 IQR (mag) 1.575 1.6 1.625 1.65 1.675 Period (days) ASAS 072851-3014.8
Fig. 7.Same as Fig.6for ASAS J072851-3014.8.
equations shown in Sect. 3.4 are strictly valid only for a pure si-nusoidal signal. In solar-like variables, the intrinsic evolution of ARs tends to modify the amplitude and phase of the rotational signal, and the use of the longer windows does not improve the precision of the period measurements. Moreover, the use of a given window tends to filter out the variability phenomena
occurring on a timescale similar to or shorter than the window length. This produces smoother curves and flattens the amplitude of the period variations. The use of a 50-d window is therefore more suitable to take into account the ARs evolution. Ideally, the length of the window should cover a small number of stel-lar rotations.Ferreira Lopes et al. (2015), for instance, suggested
12.75 13 13.25 2000 2500 3000 3500 4000 4500 5000 Median magnitude HJD-2450000 (days) -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 IQR (mag) 6.4 6.6 6.8 7 7.2 7.4 Period (days) ASAS 084229-7903.9
Fig. 8.Same as Fig.6for ASAS J0842.29-7903.9.
10.8 10.9 300 400 500 600 700 Median magnitude HJD-2450000 (days) 0 0.02 0.04 0.06 0.08 IQR (mag) 0.54 0.542 Period (days) SWASP1 J101828.70-315002.8
Fig. 9.Same as Fig.6for SWASP1 J101828.70-315002.8. The analysis with a 150-d sliding window has not been performed in this case because the two observation season spans are too short.
that long-term time series of solar-like variables should be seg-mented into sub-series of length 5 × PRotto limit the effects of
AR evolution. Unfortunately, a similar choice of window length is not feasible for most of our targets because of the sparse time-series sampling.
3.6. Comparison with other methods
In recent years, several methods have been developed to measure the SDR in solar-like stars. These methods can be grouped into
two classes. The first class comprises methods based on Doppler imaging (see e.g.Donati et al. 1997;Donati & Collier Cameron 1997;Strassmeier 2009;Waite et al. 2011), the second comprises methods based on the analysis of photometric data (see, e.g.,
Donahue et al. 1996;Reinhold et al. 2013;Lanza et al. 2014). Methods based on Doppler imaging are very effective in measur-ing the SDR in fast-rotatmeasur-ing stars (v sin i ≥ 15 km s−1) but cannot be successfully applied to slow rotators. Photometric data anal-yses can be applied to slow rotators, but only allow estimating a lower limit for the SDR. In this section, we discuss the main
differences between our method and those employed byWaite et al. (2011)andReinhold et al. (2013), which are taken as rep-resentative examples of the two classes.
3.6.1. Comparison with Zeeman Doppler imaging
Waite et al. (2011)analyzed spectropolarimetric and photometric data that were acquired during nine nights to study the topology of the magnetic field and the differential rotation of HD 106506. They complemented the information inferred from the Zeeman Doppler imaging (ZDI) with the photometric data and derived an equatorial rotation period P0= 1.39 ± 0.01 d and a photospheric
shear∆Ω = 0.21±0.03 rad d−1. They also studied the spot
distri-bution on the stellar surface and found a large polar spot coupled with low- and mid-latitude features. To compare our detection method with that employed byWaite et al. (2011), we applied our analysis to the ASAS time series of HD 106506
The shortest detected period is Pmin= 1.39 d, which is equal
to the equatorial period found byWaite et al. (2011), while the longest period is Pmax = 1.445 d. The shear inferred from Pmin
and Pmax is ∆Ωphot = 0.17 rad d−1, that is, about 20 per cent
lower than the value found byWaite et al. (2011). This difference is expected because the photometric analysis fails to detect the stellar rotation frequency at high latitudes: polar spots are always visible and do not induce an appreciable flux modulation. 3.6.2. Comparison with short-term photometry analysis
Reinhold et al. (2013) measured a lower limit of the SDR for about 20 000 late-type stars by analyzing the Q3 long-cadence photometric time-series collected by the Kepler mission. These time-series cover an interval of about 90 d and therefore are not suitable for segmentation.Reinhold et al. (2013)computed the Lomb-Scargle periodogram for the whole 90 d time series and estimated the amplitude of the SDR as
∆Ωphot= Ω1−Ω2, (7)
whereΩ1 =2πP1,Ω2 =2πP2, and P1and P2are the two most highly
significant periods detected in the periodogram. Equation (7) is based on the assumption that P1and P2 are the rotation periods
of two AR complexes located at different latitudes θ1 and θ2.
This way of measuring has two drawbacks:
– The latitude range covered by θ1and θ2depends on the phase
of the stellar magnetic cycle at which the time series was collected. If a star has a solar-like cycle, the ARs gradually migrate from high latitudes (at the minimum of cycle) to the equator (at the maximum).
– It is based on the detection of multiperiodicities in the same time series and assumes that the secondary period P2is due
to rotational modulation. However, asAigrain et al. (2015)
pointed out, the secondary peak in the Lomb-Scargle peri-odogram might also be induced by an intrinsic evolution of the ARs.
The method employed here has two advantages over the tech-nique used by Reinhold et al. (2013). First of all, it is based on the analysis of long-term photometry. The use of ASAS and Super-WASP time series permits us to study our target stars over a timescale comparable with the activity cycle duration and to track stellar rotation in a wider range of latitudes. This reduces the measurement bias that is due to the use of short-term pho-tometry. The second advantage is that our method is not based on multiple periodicities, but searches for a drift in the primary
period over the whole time series. Aigrain et al. (2015)tested different measurement methods on simulated light curves and showed that the use of multiple peaks can sometimes fail in de-tecting and measuring the SDR. The same authors point out that a more reliable SDR estimate can be achieved by searching for a drift in the mean period over an activity cycle.
4. Results
In Table 3 we report our ∆Ωphot measurements. If for a given
source both ASAS and SuperWASP estimates are available, the highest∆Ωphotvalue is reported. In the last column of the table
we indicate whether the photometric shear was computed with ASAS or SuperWASP data.
4.1. Correlation between SDR and global stellar parameters We studied the relation of our∆Ωphot measurements to the
as-trophysical parameters of our targets. The asas-trophysical parame-ters were inferred by using the infrared magnitudes MJand MH
and the theoretical isochrones developed byBaraffe et al. (1998),
Siess et al. (2000), andSpada et al. (2013). We used infrared magnitudes because the flux contrast between the spots and pho-tosphere is lower in these wavelength passbands. Hence these magnitudes are less affected by the variability induced by ro-tational modulation in comparison with the optical magnitudes. For a given star, MJ and MH are computed by adding the
dis-tance modulus DM to the J and H magnitudes reported in the 2MASS catalog. The distance modulus is computed by using the stellar parallax or the distance reported in the literature. We did not apply any correction for reddening because the difference be-tween the observed colors and those expected from the spectral type is within the photometric errors of the 2MASS photometry. The expected colors were taken from the list compiled byPecaut & Mamajek (2013)for the pre-main-sequence stars of different spectral types. In Table 4 we report the distance moduli and the inferred MJ and MHmagnitudes. In Table 5 we report for each
star the mass and effective temperature inferred from the com-parison with the different models. We also report the global con-vective turnover timescale τCinferred from the models ofSpada et al. (2013)and the derived Rossby number R=Prot
τC. The global
convective turnover timescale represents the characteristic time for the rise of a convective element through the stellar convec-tion zone(see Appendix A of Spada et al. 2013, for details on its computation).
4.1.1. Correlation between the SDR and stellar temperature In Fig.10we plot∆Ωphotvs. the color index B − V of our targets.
Different symbols are used to mark stars belonging to different associations. Although the data are quite scattered, we note a general trend with SDR amplitude increasing toward bluest col-ors. The scatter can be ascribed to two different reasons. First of all, it is related to the intrinsic limitations of our measurement method that allows only the detection of a lower limit for the SDR (see discussion in Sect. 3). The second reason is that the plot mixes stars with different ages and rotation periods. Despite the scatter, the trend shown in the picture agrees with the theo-retical models developed byKitchatinov & Rüdiger (1999)and
Küker & Rüdiger (2011). In these models, the amplitude of stel-lar SDR is inversely correlated with the depth of the convective zone, and it therefore increases toward higher temperatures.
0 0.05 0.1 0.15 0.2 0.25 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 ∆Ω phot (rad d -1 ) B - V (mag) ε Cha η Cha TWA β Pic Oct Tuc/Hor Col η Car Arg IC 2391 AB Dor
Fig. 10. Quantity ∆Ωphot measured in the present work vs. the color index B − V.∆Ωphot increases toward the bluest stars. The different symbols are used to mark stars belonging to different associations. 0.001 0.01 0.1 3500 4000 4500 5000 5500 6000 ∆Ω phot (rad d −1 ) Teff (K) HK Aqr EY Dra MS PMS Barnes et al. 2005 ∆Ω ∝ TEFF2.18 Küker & Rüdigher 2011 ∆Ω ∝ TEFF8.92 (Barnes et al. 2005) Reinhold et al. 2014
Fig. 11. ∆Ωphotvs. the effective temperature Teff. The black bullets indicate the values mea-sured in the present work. The filled bullets show stars that have reached the ZAMS, and the empty bullets show stars that are still con-tracting. The black crosses indicate the data reported inBarnes et al. (2005). The continu-ous and the dotted line depict the power law inferred in the present work and that inferred inBarnes et al. (2005). The dashed line indi-cates the power law predicted by the theoret-ical model ofKüker & Rüdiger (2011). The Teffvalues have been inferred from the models ofSiess et al. (2000). The shaded area repre-sents the uncertainty in our power-law fitting. The large gray bullets and the gray line repre-sent the results ofReinhold et al.(2013).
To quantitatively compare our results and those predicted by
Küker & Rüdiger (2011), we studied the correlation between ∆Ωphotand the stellar temperature.
In Fig. 11 we plot ∆Ωphot vs. the effective temperature
Teff inferred from the Siess et al. (2000) isochrones. The
model of Küker & Rüdiger (2011)predicts the relation∆Ω = 0.071Teff
5500
2
for ZAMS stars with Teff ≤ 6000 K. Our targets
comprise post-main sequence (PMS) stars that are still contract-ing and stars recently settled on the main sequence (MS). We selected MS stars by comparing the masses and ages of our targets with the values tabulated bySiess et al. (2000) for the ZAMS stars. We fit our ZAMS data with a power law and found ∆Ωphot= 0.09(5500Teff )2.18±0.65.
The use of Baraffe et al. (1998) or Spada et al. (2013)
isochrones does not change this result significantly. The
power laws obtained with the two models are ∆Ωphot =
0.105Teff 5500 2.06±0.7 and ∆Ωphot = 0.09 Teff 5500 2.6±0.6 , respec-tively. Although the data are quite scattered, the exponents of the theoretical and fitted power laws agree well with each other. However, the two curves exhibit a small offset of about 0.01 rad d−1and our measurements are, on average, higher than
the values predicted byKüker & Rüdiger (2011). We recall that our analysis tends to underestimate the SDR, hence the shift between theoretical and real values might be more pronounced than that shown in the figure. In Fig.11we also plot the results of Reinhold et al. (2013)and those from Barnes et al. (2005)
for comparison. We plot the median ∆Ωphot values found by Reinhold et al. (2013)in different Teff bins. These median
val-ues fall inside the 95% confidence region of our fitted power law. Hence the two works can be considered to agree. We note,
0.01 0.1 10 100 1000 ∆Ω PHOT τ C ∆Ω phot∝ τC −0.25 ± 0.04
Fig. 12. ∆Ωphotvs. the convective turnover timescale τC.
however, that the trend found byReinhold et al. (2013)is flat-ter than the power law found here and than that predicted by
Küker & Rüdiger (2011). This discrepancy may arise because
Reinhold et al. (2013)mixed stars with different ages and with a wider range of rotation periods.
Barnes et al. (2005)analyzed a sample of ten young late-type stars and found the relation∆Ω ∝ Te8.92ff . The targets investigated byBarnes et al. (2005) comprise PMS stars and stars recently settled on the ZAMS. Their power law is much steeper than that predicted by Küker & Rüdiger (2011)and disagrees with the trend of our PMS and ZAMS data. Hence our targets do not con-firm the finding ofBarnes et al. (2005).
4.1.2. Correlation between the SDR and the global convective turnover timescale
The stars we analyzed span the age range 4−95 Myr. During this time interval, the stellar structure evolves from a fully convective structure to a radiative core plus a convective envelope. In our sample, stars with the same effective temperature can therefore have very different structures depending on their age and mass. In this age range, the convective turnover timescale τCmay be
a more convenient parameter for investigating the relation be-tween the SDR and the stellar structure. Indeed, τC is
propor-tional to the depth of the convective zone and is more represen-tative of the stellar structure than of the effective temperature.
In Fig.12 we plot ∆Ωphot vs. τC. ∆Ωphot increases toward
lower τCvalues. This means that∆Ωphotincreases with
decreas-ing depth of the convective envelope. This result agrees with the theoretical models of Kitchatinov & Rüdiger (1999),Küker & Rüdiger (2011), andKüker et al. (2011). We fit a power law to our data and found∆Ωphot ∝τ−0.25±0.04C (red line). Although the
data exhibit a clear trend, they are broadly spread around the fit-ted power law. This spread is partly caused by the limitations of our measurement method and partly by the mix of stars with different rotation periods in the plot.
4.1.3. Correlation between the SDR and the rotation period In the top panel of Fig. 13we plot∆Ωphotvs. the rotation
pe-riod Prot. In this case, the data do not follow a well-defined trend
and cannot be reproduced by a power law. However, the SDR seems to be related to the stellar rotation period. The highest values of∆Ωphotcorrespond to Protbetween 0.7 and 5 d.Küker & Rüdiger (2011)studied with great detail the dependence of the SDR on the rotation period in ZAMS stars. They computed differential rotation for stars with different masses and periods and derived a set of rotational tracks. Each track shows the de-pendence of∆Ω on the rotation period for a fixed stellar mass. In the bottom panel of Fig.13we plot the rotational tracks de-rived byKüker & Rüdiger (2011)for stars with 0.5, 0.7, 0.9, and 1.1 M . According to these tracks, the SDR is almost
indepen-dent of the stellar rotation period and is more influenced by the stellar mass. We overplot our results on the tracks for compari-son. We report only stars that have reached the ZAMS because the tracks were computed for ZAMS stars. The masses of these targets cover the range 0.6−1.35 M . The circles size is
propor-tional to the stellar mass. Our results show a trend that partly resembles the rotational tracks. The less massive stars exhibit on average a lower SDR. However, stars with Protbetween 0.7 and
2 d diverge strongly from the model predictions. This suggests that the stellar mass cannot be the main parameter on which the SDR depends, as claimed byKüker & Rüdiger (2011), and that stellar rotation period plays a key role as well.
4.1.4. Correlation between the SDR and stellar age
The stellar associations we studied span an age range between 4 and 95 Myr. Hence our sample includes stars that are in the first phases of the PMS stage and stars that are approaching or have recently settled on the MS. Our results are therefore useful to investigate how the SDR evolves in time during the PMS stage. In the top panel of Fig.14we display the measured values of ∆Ωphotvs. the stellar age. The median∆Ωphotincreases between
4 and 30 Myr and then decreases and approaches the current so-lar value at 95 Myr. The scatter of the data is partly caused by the intrinsic limitations related to our methodology and partly by the mixed stellar masses and rotation periods. In the bottom panel of Fig. 14we report only stars with masses between 0.85 and 1.15 M and investigate how∆Ωphotevolves in a young Sun. The
0 0.05 0.1 0.15 0.2 0.25 0 5 10 15 20 25 ∆Ω phot (rad d −1 ) ε Cha η Cha TWA β Pic Oct Tuc/Hor Col η Car Arg IC 2391 AB Dor 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 2 4 6 8 10 ∆Ω phot (rad d −1 ) Prot (d) MS 0.5 M⊙ 0.7 M⊙ 0.9 M⊙ 1.1 M⊙
Fig. 13. Top panel:∆Ωphotvs. the stellar rotation period Prot. The highest∆Ωphot val-ues have been measured in stars with Prot≤ 5d. Bottom panel: the same plot restricted to the MS stars. The symbol sizes are pro-portional to the stellar masses, which range between 0.6 and 1.35 M . The black lines are the rotational tracks derived byKüker & Rüdiger (2011)for stars of 0.5, 0.7, 0.9, and 1.1 M as labeled. 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 ∆Ω phot (rad d −1 ) All stars ε Cha η Cha TWA β Pic Oct Tuc/Hor COL η Car Arg IC 2391 AB Dor Solar value 0 0.05 0.1 0.15 0.2 0.25 0 20 40 60 80 100 ∆Ω phot (rad d −1 ) Age (Myr) 0.85 M⊙ < M < 1.15 M⊙ Median Küker & Stix (2001)
Fig. 14. Top panel: ∆Ωphot vs. stellar age. The empty circles indicate the median ∆Ωphot values measured at different ages (note that the members of η Cha have been grouped with the members of TWA). The black continuous line connects the median ∆Ωphotvalues and the dotted line marks the present solar shear for comparison. Bottom panel: the same plot restricted to stars with mass between 0.9 and 1.1 M . The gray crosses mark the rotational shear for the Sun at different ages, as predicted byKüker & Stix (2001).
increases, and at 30 Myr, its value is about twice the initial value. Between 30 and 95 Myr∆Ωphotremains almost constant.Küker & Stix (2001)modeled the SDR evolution in the Sun. They de-veloped four theoretical models for the Sun at 3, 10, and 31 Myr and for the present Sun. In their models ∆Ω is inversely cor-related to the depth of the convective zone. This implies that ∆Ω increases between 3 and 31 Myr. At 3 Myr, the Sun has a fully convective structure and a low SDR. As the Sun evolves in time, a radiative core grows, the depth of the convective zone decreases, and∆Ω increases.Küker & Stix (2001)did not find a significant difference between the 31 Myr and the present Sun models. Hence, according to their work, the Sun should have reached its current∆Ω value at about 30 Myr. In this case, our results also agree only qualitatively with the model prediction. The median∆Ωphotvalues increase between 4 and 30 Myr, but
they are systematically higher than those predicted byKüker & Stix (2001). We note that although these young stars exhibit an SDR greater than solar, they can be regarded as solid-body ro-tators because they rotate faster than the present Sun and have a low relative shear α=Ω∆Ω
eq.
5. Discussion
In the previous section we showed that our results agree qual-itatively with the models developed byKitchatinov & Rüdiger (1999)andKüker & Rüdiger (2011). However, our∆Ωphot
mea-surements are often higher than the values predicted by the same models (see Figs.11and13), especially in stars with periods be-tween 0.7 and 2 d. This disagreement bebe-tween∆Ω values and model predictions has previously been noted and discussed by
0 0.05 0.1 0.15 0.2 0.25 0.001 0.01 0.1 1 ∆Ω phot (rad d -1) ε Cha η Cha TWA β Pic Oct Tuc/Hor Col η Car Arg IC 2391 AB Dor 0 0.05 0.1 0.15 0.2 0.25 0.3 0.001 0.01 0.1 1 αphot Rossby nr.
Fig. 15. Top panel: absolute shear∆Ω vs. the Rossby number Ro. Bottom panel: the rela-tive shear α vs. Ro.∆Ω and α increase toward higher Ro values. This trend agrees with the model developed byGastine et al. (2014).
Marsden et al. (2006). These authors measured an absolute shear ∆Ω = 0.4 rad d−1for the G0 star HD 171488. This value is
sig-nificantly higher than those measured for the G dwarfs R58 and LQ Lup, which are∆Ω = 0.138 rad d−1and∆Ω = 0.12 rad d−1, respectively(Marsden et al. 2005; Donati et al. 2000). According to the models, the three stars should have about the same rota-tional shear because the stellar temperature is the main param-eter on which∆Ω depends.Küker et al. (2011)were unable to explain this discrepancy and questioned the reliability of the∆Ω measurement for HD 171488. However,Jeffers & Donati (2008)
made an independent measurement and found∆Ω = 0.5 rad d−1 The main difference between HD 171488 and the other two stars lies in the rotation period. HD 171488 has a rotation period of 1.31 d, while the other two stars have a rotation period of 0.5 and 0.3 d, respectively.
The rotational tracks computed byKüker & Rüdiger (2011)
predict that the SDR amplitude slightly increases with the period in the range 0.3−2 d (see bottom panel of Fig.13), but this incre-ment is too small to explain the SDR amplitude of HD 171488. The high differential rotation found in HD 171488 and in our targets, with periods between 0.7 and 2 d, suggests that the de-pendence on the rotation period could be more pronounced than models prediction.
A stronger dependence on the stellar rotation rate could also explain the discrepancy between the power law found byBarnes et al. (2005)and that found in the present work (see Fig.11). The high steepness of the power law found byBarnes et al. (2005)is due to the M1V stars HK Aqr and EY Dra, for which the authors measured very low SDR values of 0.005 and 0.0003 rad d−1, respectively. The M1V stars of our sample have higher∆Ωphot
values. However, we point out that HK Aqr and EY Dra rotate in about 0.5 d, while the M1V stars investigated here have rotation periods of between 1.5 and 5.5 d. The difference between HK Aqr, EY Dra, and the M1V stars investigated here might there-fore be due to the different rotation periods.
The models developed byKüker & Rüdiger (2011)are based on the so-called Λ effect. In these models the main driver of the differential rotation is the non-diffusive term of the Reynolds stress induced by the interaction between the Coriolis force and
the convective motions. As noted byGastine et al. (2014), these models depend on free parameters such as the turbulent viscosity coefficients. A different choice of these parameters might lead to a stronger dependence on stellar rotation rate.Gastine et al. (2014)developed a model based on a 3D hydrodynamical code where the Reynolds stresses do not need to be parametrized. In this model, the main parameter on which∆Ω depends is the Rossby number Ro=Prot
τc , where Protis the stellar rotation period
and τcis the convective turnover timescale. If Ro < 1, which is
the case of our targets, the Coriolis force dominates the buoyancy and the star tends to rotate as a solid body. As Ro increases, the Coriolis force becomes weaker and the value of differential rota-tion increases. For Ro > 1 the buoyancy dominates on Coriolis force and the SDR becomes antisolar. In Fig.15we plot∆Ωphot
vs. the Rossby number computed in Sect. 3. The plot is quite scattered, but the trend of∆Ωphotseems to confirm the
predic-tion ofGastine et al. (2014). In the bottom panel of Fig.15we report the relative shear αphotvs. the Rossby number. The trend
and the range of αphotvalues are very similar to that predicted by Gastine et al. (2014), which is shown in their Fig.2.
6. Conclusions
We investigated the correlation between the SDR amplitude and global stellar parameters in members of young loose stel-lar associations. We measured the quantities ∆Ωphot and αphot
for 111 stars by processing long-term photometric time-series. These quantities are lower limits to the absolute and to the rel-ative surface rotational shear. Our analysis led to the following results:
– ∆Ωphot increases with effective temperature Teff following
the power law ∆Ωphot ∝ Te2.18±0.65ff in MS stars. This power
law is very close to that predicted byKüker & Rüdiger(2011; i.e.,∆Ω ∝ Te2ff).
– The PMS stars of our sample show a trend very similar to that exhibited by MS stars. Thus, the power law∆Ω ∝ T8.6 eff
found byBarnes et al. (2005)for PMS and ZAMS stars is not confirmed by our measurements.
– ∆Ωphot increases with decreasing convective turnover
timescale τCaccording to the power law∆Ωphot∝τ−0.25±0.4C .
– Our∆Ωphotmeasurements are systematically higher than the
values predicted byKüker & Rüdiger (2011). This discrep-ancy is particularly large in stars with a rotation period of between 0.7 and 2 d and suggests that the dependence on the rotation period may be stronger than the model prediction. – We investigated the time evolution of∆Ωphotfor a 1 M stars
and found that∆Ωphotincreases with stellar age in the first
30 Myr. This is consistent with the theoretical models that predict a low degree of differential rotation for fully convec-tive stars.
– ∆Ωphot and αphot increase with the Rossby number Ro in
agreement with the theoretical model developed byGastine et al. (2014).
Acknowledgements. The extensive use of the SIMBAD and ADS databases op-erated by the CDS centre, Strasbourg, France, is gratefully acknowledged. We used data from the WASP public archive in this research. The WASP consor-tium comprises of the University of Cambridge, Keele University, University of Leicester, The Open University, The Queen’s University Belfast, St. Andrews University, and the Isaac Newton Group. Funding for WASP comes from the consortium universities and from the UK’s Science and Technology Facilities Council. The Authors would like to thank Dr. G. Pojmanski for the extensive use we made of the ASAS database. We are grateful to Rainer Arlt and to the anonymous referee for helpful comments and suggestions.
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